Calculation of the Ettingshausen coefficient in quantum wells with parabolic potential in the presence of electromagnetic wave (for electron-confined acoustic phonons scattering)

The analytic results have shown that EC depends on temperature, magnetic field, characteristic quantities of EMW and m - quantum number which is specific the confined phonons in a complicated way. The numerical results for GaAs/GaAsAl quantum wells (QW) have displayed these dependence explicitly. In particular, when m is set to zero, we achieve results for magneto – thermoelectric effect in the same QW without the confinement of acoustic phonons.

TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO

ISSN: 2354 - 1431 http://tckh.daihoctantrao.edu.vn/

Calculation of the Ettingshausen coefficient in quantum wells with parabolic potential in the presence of electromagnetic wave (for electron-confined acoustic phonons scattering)

Nguyen Thi Lam Quynha*, Nguyen Ba Ducb, Nguyen Quang Baua

a

VNU University of Science

b

Tan Trao University

*

Email:lamquynh.katty@gmail.com

Recieved:

28/8/2018

Accepted:

10/9/2018

By using the quantum kinetic equation for the distribution function of electrons, the expression for Ettingshausen coefficient (EC) in quantum wells with parabolic potential (QWPP) in the presence of electromagnetic wave (EMW) is obtained for electrons - confined acoustic phonons scattering The analytic results have shown that EC depends on temperature, magnetic field, characteristic quantities of EMW and m - quantum number which is specific the confined phonons in a complicated way The numerical results for GaAs/GaAsAl quantum wells (QW) have displayed these dependence explicitly In particular, when m is set to zero, we achieve results for magneto – thermoelectric effect in the same QW without the confinement of acoustic phonons

Keywords:

quantum wells,

Ettingshausen efffect,

magneto – thermoelectric

effect, quantum kinetic

equation, confined

acoustic phonons.

1 Introduction

Both wave function and energy spectrum of the

electrons are quantized under the influence of

confinement effect So, the low-dimensional

semiconductor systems (LDSS) have not only changed

physical properties but also being appeared new

effects [1-5] Among them, we have to mention

Ettingshausen effect That is a thermoelectric

phenomenal that effects the current in conductor in the

presence of magnetic field The creation of

electronhole pairs at one side and their recombination

at the other side of the sample are the main cause of

Ettingshausen effect in semiconductors [6] This effect

was also studied some twodimensional semiconductor

systems [3,4] However, those studies have not

interested in the confinement of phonons In other

effects in LDSS: confined LO-phonons create new properties of the Hall effect in doped semiconductor supperlatices [1]; confined optical phonons makes a remarkable impact on the Hall effect [2] and increase the number of resonance peaks of the nonlinear absorption coefficient of a strong electromagnetic wave by confined electrons [5] in a compositional supperlatices So far, how the CAP influence on the Ettingshausen effect in QWPP is still an unanswered question

In this work, a QWPP in the presence of constant electric field, magnetic field and EWM have been considered for Ettingshausen effect [3] We have taken electron-CAP scattering into account and obtained analytic expression for the EC In the process

of transformation, we always count on the temperature

Components of the article are as follows: In

section 2, we get the analytic equation of the EC based

on computation related to the Hamiltonian of electron

We give the result of numerical calculation and

discussion in section 3 Final section contains

conclusions

2 The Ettingshausen coefficient in the QWPP

under the influence of confined acoustic phonons

We have considered a QW with parabolic

potential:

2 2 ( )

2

z

V  m w (with w is detention z

frequency characteristic QWPP).There exists a

magnetic field with B   0, 0, B 



and constant

electric field withE1  E1, 0, 0 



In this case, the movement of electrons is limited to Oz; so, they can

only move freely in the x-y plane with cyclotron

frequency c

e

eB w

m

 and imply velocity 1

d

E v B



That means QWPP have been considered in the

condition: the magnetic field is perpendicular to the

free-moving plane of electrons Energy of an electron

is and being received intermittent values:

(2.1)

Here   py

is the wave vector of electrons in the

y-direction

When QWPP is subjected to a laser radiation

 

Hamiltonian of the electron CAP system can be expressed as:

2.2)

In which:

, , , , ,

N n p N n p

a  a   , , , 

m q m q







  are the creation and annihilation operators of electrons

(phonons) respectively;   0cos Ω  

Ω

c



is

the vector potential of laser field;

q

  

 

 is scalar potentialwith unit vector in the direction of magnetic field H

h H







;

m q

m π

L





energy of a CAP with the wave vector q   q q, z

  

and q qx qy

  

; m is the detention index of phonons

m

with  

2

z m

s

ξ q q

C q

ρv









 

is the electron -

CAP interaction constant ( , ,ξ ρ v are s the deformation potential constant, the mass density and the sound velocity, respectively)

is the electron form factor

withLN NN  u

is the associated Laguerre polynomial

The quantum kinetic equation of average number

of electron is:

in which

Using (2.2) for (2.3) then we performed

transformations of operator algebra and obtained:

where:

0

Ω

y

e

eE q

λ

m





;

,

, m q ,



is the equilibrium distribution function of the

phonons

For simplicity, we limit to the case ofl  0, 1  ,

get to close

We multiply both sides by (2.4) with

 

e

p δ ε ε p

then taking sum of N, n, and

y

p

 

We get following expression:

(2.5)

In the above expression, we use symbols to replace

complex equations    h G ε ,     

 

directional

multiplication of h

and G ε  



(2.6) And

withτ is the momentum relaxation time and 1

F

ε ε

T







(ε F is the Fermi energy of

electron)

By solving the equation (2.5), we find out expression of individual current density:

(2.8)

The total current density J

and the thermal flux

density Q



are given by:

(2.9)

And

(2.10)

In low temperature conditions, the electron gas in

QW is completely degenerated The equilibrium distribution function of electron is of the form:

0

0

f   n θ ε  ε  The distribution function of electron is found in linear approximation by:

(2.11)

here:

(2.12)

From expressions of the total current density and the thermal flux density achieved, comparing it to the writing:Jp σ Eip 1p β Tip and Q μ E φ Tp ip 1p ip

we obtain analytic expression of tensors:

Here:

With

u

c

m

ω



with

u

c

ω





ijk

λ is the anti-symmetrical Levi tensor; δ kpis the Kronecker delta and i, j, k, l, p correspond the components x, y, z of the Cartesian coordinates The expression of the EC is given by:

(2.17)

In Eq.(2.17),

are components of tensors in Eq.(2.13), Eq.(2.14), Eq.(2.15) and Eq.(2.16), respectively; KL is the thermal conductivity of phonons From analytic expressions, we can see that the EC depends in a complicated way on characteristic quantities of EMW (the amplitude E0 and the frequency Ω), the temperature, the magnetic field, and especially the m-quantum number being specific to the confined phonons Interesting the energy of CAP

m π

L







  leads to abundant analytic results

and being added to resonance condition in QW In particular, we get the results in the case of unconfined acoustic phonons when m is set to zero [3] These dependencies will be clarified in section 3 when we study QWPP of GaAs/GaAsAl

3.Numerical results and discussions

To get influence of the CAP on the EC in QWPP

in the presence of EMW in detail, we consider the QWPP of GaAs/GaAsAl with the parameters:

0 0.067 e

m  m (m eis the mass of a free electron),

1

electron’s detention index (n, n’, N, N’) rate from 1

to 3

Figure1 The dependence of the EC on EMW

amplitude

Fig.1 describes the dependence of EC on EMW

amplitude in two cases: with and without confinement

of acoustic phonons at T=5K The graph indicates

that: the EC depends clearly on the EMW in low

amplitude domain The EC rises fast and linearly to

reach the horizontal line in both cases to be considered

in higher amplitude region We realize that in the high

EMW amplitude condition, the EC is almost

unchanged when the EMW amplitude increases

Besides, the EC has negative values with unconfined

phonons [3] and even confined

As can be seen from Fig.2, the EC oscillates

strongly when the EMW frequency is less than

12

10 Hz When the EMW frequency increases from

12

10 Hzto 2,0.10 Hz12 the EC has the same value and

almost be unchanged in both cases In this frequency

range, both EC peaks and EC peak positions tend

upward The graph also shows that: peaks of the blue

line are sideways to the right and be higher than peaks

of red line We can explain those results as follows:

the resonance peaks correspond to the condition:

or

; so, when m increases,the resonance peaks tend to

shift to higher frequency regions and corresponding to

each resonant frequency, the EC has greater value

Meanwhile, the EC always increases when the EMW

frequency increases in the same frequency domain as

in electron optical phonons scattering [4] Moreover,

in the case of electron acoustic phonons scattering, the

EC has negative values This result is completely

opposite to case of electron optical phonons scattering

the EC has positive values [4] Thus, the scattering

mechanism not only affects the values but also the

variation of the EC under influence of EMW

frequency change

Figure 2 The dependence of the EC on EMW

frequency

Figure 3 The dependence of the EC on

temperature Fig.3a indicates that in both cases - with and without the confinement of acoustic phonons - the EC has negative values and be nearly linear when the temperature increases In particular, when m goes to zero we obtain the results in the same QWPP in the case of unconfined acoustic phonons [3]

The influence of EMW on the EC is displayed clearly in the Fig.3b In the temperature domain investigated, the EC has greater values within the presence of the EMW and the confinement of acoustic

only causes change in the magnitude of the EC while

temperature increases

In the Fig.4a, we can see oscillations of the EC

when magnetic field changes The graph shows that

both lines oscillate and reach resonant point The blue

line (with CAP) not only has more resonance peaks

than the red line (without the confinement of acoustic

phonons) but peaks of the blue line are also taller than

the red line’s We can easily explain as follows: when

acoustic phonons are confined, their wave vector is

quantizied; both energy and interaction constant

depend on quantum number m; so, the resonance

condition is affected by m: the larger the value of m

received, the more the resonance peaks of EC That

means the confinement of acoustic phonons affect the

EC’s changing law under increasing of magnetic field

Figure 4 The dependence of the EC on magnetic

field

The existence of EMW also governs the EC’s law

of change It is displayed in Fig.4b E0 is appeared in

the argument of the Bessel function and not related to

the resonance condition When E 0 0resonance

peaks are sideways to the left and have greater values

in comparison to the case of 5 

0 4 1 0 /

These are different from the case of unconfined acoustic phonons [3]

4.Conclusions

By using the quantum kinetic equation for electron with the presence of invariable electric field, magnetic field and EMW, in this paper, we have calculated the analytic expression of the EC, graphed the theoretical results for GaAs/GaAsAl QWPP The achievements get show that the formula

of EC depends on many quantities, especially the quantum index m specific the confinement of phonons All of numerical results indicate that the quantum number m have impacted to the EC The

EC values are greater when we carry out the survey within confinement of acoustic phonons When acoustic phonons are confined, the EC values or absolute values of the EC are 6 to 10 times as much

as the EC without confinement of phonons In addition, the m also affects the resonance condition and makes the appearance of auxiliary resonance If

m goes to zero, the results obtained come back to the case of unconfined phonons and ignored the energy of acoustic phonons [3] In the comparison with the case of electron–optical phonons scattering [4], a few results we achieved which are completely opposite That means the scattering mechanism not only affects the values but also the variation of the

EC Finally, we can assert that the confinement of acoustic phonons creates surprising changes of the

EC in the QWPP

Acknowledgments

This work was completed with financial support from the National Foundation for Science and Technology Development of Vietnam (103.01-2015.22)

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Impact of confined LO-phonons on the Hall effect in doped semiconductor supperlatices, Journal of

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149-154;

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Minh Quang and Nguyen Thi Thanh Nhan

(2017),Magneto-thermoelectric effect in quantum well

in the presence of electromagnetic wave, VNU Journal

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Thanh and Nguyen Quang Bau (2016),The

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Quang Bau (2012), The impact of confined phonons

on the nolinear obsorption coefficient of a strong electromagnetic wave by confined electrons in compositional supperlatices, VNU Journal of Science,

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Tính toán hệ số Ettingshausen trong hố lượng tử thế parabolkhi có mặt sóng điện

từ (trường hợp tán xạ điện tử-phonon âm giam cầm)

Nguyễn Thị Lâm Quỳnh, Nguyễn Bá Đức, Nguyễn Quang Báu

Thông tin bài viết Tóm tắt

Ngày nhận bài:

28/8/2018

Ngày duyệt đăng:

10/9/2018

Biểu thức của hệ số Ettingshausen trong hố lượng tử với hố thế parabol khi có sóng điện từ được thu nhận trên cơ sở phương trình động lượng tử cho hàm phân bố của điện tử trong trường hợp tán xạ điện tử - phonon âm giam cầm Các kết quả giải tích đã chỉ ra sự phụ thuộc phức tạp của hệ số Ettingshausen vào nhiệt độ, từ trường, các đại lượng đặc trưng của sóng điện từ và số lượng

tử m đặc trưng cho phonon giam cầm Những sự phụ thuộc này được hiển thị

rõ nét trong kết quả tính toán số cho hố lượng tử GaAs/GaAsAl Đặc biệt, khi cho m tiến về không, ta thu được kết quả của hiệu ứng từ-nhiệt-điện tương ứng với trường hợp phonon không giam cầm trong hố lượng tử cùng loại

Từ khoá:

Hố lượng tử, hiệu ứng

Ettingshausen, hiệu ứng

từ-nhiệt-điện, phương trình

động lượng tử, phonon âm

giam cầm